Stable System

Definition

A stable system is one that, when subjected to bounded input, produces a bounded output over time. This concept is crucial for understanding the behavior of linear time-invariant (LTI) systems, as stability ensures that the system will respond predictably and not produce unbounded responses, which can be critical in applications such as control systems and signal processing.

5 Must Know Facts For Your Next Test

1. For an LTI system to be stable, all poles of its transfer function must lie within the unit circle in the complex plane.
2. A stable system will return to equilibrium after being disturbed by an external input or disturbance.
3. Stability can be analyzed using tools like the Routh-Hurwitz criterion or Nyquist criterion, which help determine if a system remains stable across various conditions.
4. In causal systems, stability implies that the impulse response must be absolutely summable or square summable.
5. For discrete-time systems, BIBO (Bounded Input Bounded Output) stability is a key concept, meaning if the input is bounded, the output will also be bounded.

Review Questions

• How does stability relate to causality in LTI systems?
  • Stability and causality are closely related properties in LTI systems. A causal system only depends on present and past inputs, which means its output will not 'anticipate' future inputs. For such systems to remain stable, they must respond within certain limits when subjected to inputs; otherwise, they risk producing unbounded outputs over time. Thus, while causality ensures that a system's response is timely, stability ensures that this response remains predictable and bounded.
• Explain how you can determine if a discrete-time LTI system is stable using pole-zero analysis.
  • To determine if a discrete-time LTI system is stable through pole-zero analysis, you need to examine its transfer function represented as a ratio of polynomials. The poles of this transfer function are found by setting the denominator equal to zero. A system is stable if all poles are located inside the unit circle in the complex plane. This visual approach allows you to quickly assess stability by plotting the poles on a pole-zero plot and verifying their positions relative to the unit circle.
• Evaluate the implications of instability in a bioengineering context when designing feedback control systems for medical devices.
  • Instability in feedback control systems for medical devices can have dire consequences, potentially leading to erratic behavior or failure to maintain critical physiological parameters. In bioengineering applications, ensuring that systems are stable means they can effectively monitor and respond to changes in patient conditions without causing undue fluctuations or risks. The consequences of unstable systems could range from ineffective treatments to severe patient harm, highlighting the importance of thorough stability analysis during the design phase of any medical device.

Related terms

Causality: A property of a system where the output depends only on present and past inputs, ensuring that no output occurs before an input is applied.

Impulse Response: The output of a system when an impulse function is applied as input, used to analyze the stability and behavior of the system.

Pole-Zero Plot: A graphical representation of the poles and zeros of a system's transfer function, which helps to determine the stability and frequency response of the system.